Some new results for subsequences of N\"orlund logarithmic means of Walsh-Fourier series

We prove that there exists a martingale $f\in H_p$ such that the subsequence $\{L_{2^n}f\}$ of N"orlund logarithmic means with respect to the Walsh system are not bounded in the Lebesgue space $weak-L_p$ for $0<p<1$. Moreover, we prove that for any $f\in L_p(G),$ $p\ge 1,$ $L_{2^n}f$ converge to $f$ at any Lebesgue point $x$. Some new related inequalities are derived.